Approximation by smooth multivariate splines
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The degree of approximation achievable by piecewise polynomial functions of given total order on certain regular grids in the plane is shown to be adversely affected by smoothness requirements-in stark contrast to the univariate situation. For a rectangular grid, and for the triangular grid derived from it by adding all northeast diagonals, the maximum degree of approximation (as the grid size 1 /n goes to zero) to a suitably smooth function is shown to be O(n-ρ-2) in case we insist that the approximating functions are in Cρ. This only holds as long as (FORMULA PRESENTED), respectively, with r the total order of the polynomial pieces. In the contrary case, some smooth functions are not approximate at all. In the discussion of the second mesh, a new and promising kind of multivariate B-spline is introduced. © 1984 American Mathematical Society.
author list (cited authors)
de Boor, C., & DeVore, R.